Kinematics of rotation
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In rotational motion, we cannot treat the object as a particle because at any given time different parts have different linear velocities and linear accelerations. The angular displacement (thus the angular frequency and the angular acceleration) will be the same for all particles of the rigid object because the relative locations of all particles remain constant. This means we can define the angular quantities for the object as a whole. These angular quantities can be used the same way as in circular motion. We can speak of uniform rotation and rotation with constant angular acceleration; the corresponding formulae are the same as those listed in circular motion. The only significant difference is that it is sometimes convenient to use an angular velocity vector and an angular acceleration vector. |
Angular velocity vector
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Angular velocity (\(\mathbf{ω}\)) is a vector whose magnitude is equal to the angular speed and whose direction lies in the axis of rotation, depending on the direction of the rotation as determined by the right-hand rule. If the fingers on your right hand curl in the direction of the rotation, your thumb points in the direction of the angular velocity. If the axis of rotation stays constant (eg, a door on its hinges), the direction of the angular velocity remains constant (the magnitude may change). If the axis of rotation changes (eg, a spinning top), the direction of the angular velocity also changes. The relationship between the linear velocity \(\mathbf{v}\) of a point of the rotating rigid body at position \(\mathbf{r}\) (remember the relationship \(v = \omega r\) for the angular speed): \[\mathbf{v} = \mathbf{ω} \times \mathbf{r}.\] |
Angular acceleration vector
The angular acceleration vector \(\mathbf{α}\) is the rate of change of the angular velocity vector. This change may include change in magnitude or in direction.
\[\mathbf{α} := \lim\limits_{\Delta \to 0}\frac{\Delta \mathbf{ω}}{\Delta t} = \frac{\mathrm{d}\mathbf{ω}}{\mathrm{d}t}.\]